Performance of hollow steel tube bollards under quasi-staticand lateral impact load

Soheila Maduliat, Tuan Duc Ngo n, Phuong Tran, Raymond LumantarnaDepartment of Infrastructure Engineering, The University of Melbourne, Melbourne, VIC 3010, Australia

a r t i c l e i n f o

Article history:Received 23 July 2014Received in revised form25 November 2014Accepted 26 November 2014

Keywords:Hollow steel tubesCollapse behaviourYield line mechanismsEnergy absorption

a b s t r a c t

In order to prevent vehicle access to a protected area, vehicle barriers can be installed around theperimeter of the area. Bollards are commonly used as vehicle barriers. This is due to the fact that theycan be readily blended with other architectural features and present fewer disturbances to a building’sfunctionality when compared to other barrier systems. Hollow steel tubes are used in a variety of barriersystem applications where they are required to absorb deformation energy. Varying methods, such asfinite element analysis or experimental observation, can be used to determine the collapse behaviourand energy absorption of these steel structures under lateral impact load. These methods have highaccuracy but demand a significant amount of time and computational resources. Apart from experi-mental and numerical analyses, Yield Line Mechanism (YLM) is an approach that can provide thecollapse response of sections. This is when a section fails and the YLM of failure forms at its localisedplastic hinge point. The YLM analysis approach is commonly used to investigate the performance of thin-wall structures that have local failure mechanisms. This paper investigates the collapse behaviour andenergy absorption capability of hollow steel tubes under large deformation due to lateral impact load.The YLM technique is applied using the energy method, and is based upon measured spatial plasticcollapse mechanisms from experiments. Analytical solutions for the collapse curve and in-plane rotationcapacity are developed, and are used to model the large deformation behaviour and energy absorption.The analytical results are shown to compare well with the experimental values. The YLM model is thenused to verify the finite element model (FEM), and then the failure behaviour and energy absorption ofhollow steel tubes under lateral impact load is investigated in more detail.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Steel hollow sections have been used as structural elements inbuilding systems and highway barriers, and may be required todissipate energy when subjected to dynamic loads. They areprimarily made from thin-walled sections due to these sectionsbeing efficient and versatile (Nagel [19]). Steel sections areeconomical and provide valuable mechanical properties for usein industry; however, they are prone to local failure due to theirthin-walled elements. Thus, the deformation of these sectionsneeds to be investigated in order to assess the performance ofsteel sections against impact loading.

The performance of hollow sections subjected to axial impactload have been thoroughly investigated (Boutros [6]; Boutros et al.[7]; Daneshi and Hosseinipour [8]; Hosseinipour [12]; Jang et al.[13]; Adachi et al. [3]; Zhang et al. [24,25]; and Alavinia et al. [2]).However, the behaviour of hollow sections subjected to lateral

impact loading has not been clearly established, whilst theperformance of bollard systems is determined based on fieldtesting standardised in PAS 68 [21]. In this study, the behaviourof steel tube sections subjected to a combination of local deforma-tion due to lateral impact load and bending failure is studied.

When thin-walled sections fail due to lateral impact load, theyundergo plastic folding of the cross-section walls. Yield line mechan-ism (YLM) analysis of the collapse mechanism provides a bendingmoment–rotation relationship, fromwhich the member strength andenergy absorption capacity can be estimated. In the elastic range,where the deformations of the elements are small, the theory ofelasticity can be used to determine the load-deformation behaviourof the structure. When increasing the load, local yielding occurs andhinges may develop. The collapse behaviour of the element dependson the behaviour of the plastic hinges. Failure mechanism (yield linemechanism) theory can be used to determine the load deformationbehaviour of the structure in the post-failure range.

There are different theories to analyse the collapse behaviour ofa complete structure. However, for achieving correct results from atheory, an accurate model should be prepared. The YLM modelsare based on experimental observations.

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Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2014.11.0240263-8231/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ61 3 83447950; fax: þ61 3 83444616.E-mail address: dtngo@unimelb.edu.au (T.D. Ngo).

Thin-Walled Structures 88 (2015) 41–47

Based on laboratory test observations, Murray and Khoo [18]developed eight basic mechanisms for plates and five combina-tions of simple mechanisms for channel columns.

Kecman [15] studied the bending collapse behaviour of rectan-gular and square hollow sections and subsequently developed a YLMmodel, including travelling yield lines. Koteko [16] investigated theYLM of rectangular and trapezoidal box section beams with a highwidth to depth ratio compared to Kecman’s sections. In Elchalakaniet al. [10], the collapse mechanism model for tube members wasdiscussed and included the effect of ovalisation along the length ofthe tube. According to this study, the mechanism starts when the

major axis of the oval reaches 2.2R and the minor axis reaches 1.8R,where R is the radius of the section (Fig. 1).

The mechanisms presented in this paper are based on theexperimental observations for hollow sections from studies con-ducted by Elchalakani et al. [10] and Poonaya et al. [20]. The plasticcollapse curves developed from the analysis are shown to comparewell with the experimental curves. To fully describe the largedeformation of the sections, collapse curve solutions are used inconjunction with elastic and in-plane plastic theory. By using thebending moment–rotation curve, the total absorbed energy can becalculated analytically. The energy absorption results are shown tocompare well with the experimental values, and provide analysiswith a robust tool for estimating the energy absorption capacity ofsteel hollow sections under lateral impact load.

2. Development of the YLM model for hollow sections

Experimental observations are used to define a basic YLMmodel. Fig. 2 shows the common failure mode of steel hollowsections under lateral impact load. It is noted that the collapsedeformation shape of the bollards under lateral impact load aresimilar to the observed deformation shape described in theresearch work by Poonaya et al. [20] (Fig. 3). In their study, thecollapse curve and ultimate capacity of the thin-walled tube underbending was investigated experimentally.

Since the deformation shapes of steel hollow sections underlateral impact load and bending are the same, this study uses theexperimental results based on Elchalakani et al. [10] and Poonayaet al. [20] to validate the YLM.

By establishing the YLM model for the sections and calculatingthe energy absorption of each hinge line, the total absorbed energycan be estimated and therefore, a failure curve can be plotted. Thefollowing section explains the calculation of total energy absorp-tion of a defined model for different rotation angles in order toplot the failure curve.

2.1. Failure curve

The energy method is used to estimate the failure curve of thehollow section. The total energy absorption for the YLM model is

Fig. 1. (a) Front view of deformed section (Poonaya et al. [20]), (b) ovalisation ofhollow steel tube.

Local deformation of bollard due to impact load

Global deformation of bollard due to impact load

Experimental observation to define a basic YLM model

Fig. 2. Failure mode of steel hollow section.

S. Maduliat et al. / Thin-Walled Structures 88 (2015) 41–4742

the sum of each individual hinge line work, as follows:

W θ� �¼ ∑

n

1WiðθÞ; ð1Þ

where n is the number of hinge lines in the model. The associatedbending moment is defined as:

MðθÞ ¼ dWdθ

; ð2Þ

where θ is the rotation angle of the beam. Therefore, for differentvalues of the rotation angle, the bending moment can be deter-mined and the failure curve (moment–rotation graph) can beplotted. The following paragraphs explain the calculation of eachindividual hinge line work that is proposed with the YLM model.

The energy absorptions for the compression hinges are calcu-lated based on the work components defined by Kecman [15]. Allof the work components are shown in Fig. 4.

Fig. 4 contains eleven plastic hinge lines associated with lines:BB1, CF, CE, AF, AE, AB, AB1, BF, B1F, BE and B1E.

These work components for each plastic hinge (W1,W2,W3,W4,W5 and W6) are defined as follows:

W1 ¼WBB1 ¼mpLBB1 ð2γÞ; ð3Þ

where WBB1 is the work component to form a hinge line betweenpoints B and B1; γ is the rotation angle due to formation of a hinge

line between points B and B1 and can be determined usingEq. (3c); mp is the plastic moment capacity of the steel elementsand is determined using Eq. (8); and LBB1 is the length of the hingeline between points B and B1, which can be calculated as follow:

LBB1 ¼ Rðπ�2αÞ; ð3aÞwhere R is the radius of the steel hollow section and is shown onFig. 1.

α¼ arcsin0:8RR

� �; Refer to Fig: 1 ð3bÞ

γ ¼ arccosa1�2R tan θ� �

cos θ� �

a2

� ð3cÞ

a2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21þð0:5LBB1 Þ2

� �r; a1 ¼

2R3

ð3dÞ

Similarly,

W2 ¼WCFþWCE ¼ 2mpa1π4�α2

� �ð4Þ

Here, π=4� �� α=2

� �� �is the rotation angle of hinge lines CF and

CE after failure and is shown on Fig. 1.WAE is equal to WAF; therefore we have derivation for the 3rd

energy component as:

W3 ¼ 2ðWAEþWAF Þ ¼ 4mpη1LAE ð5ÞHere,

LAE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22þð0:5LBB1 Þ2

qð5aÞ

η1 is the angle between surface AED and AEB (∠AED; AEB) due todeformation of the steel hollow section, and is determined by thefollowing equations:

η1 ¼ arctanLBDLDA''

0@

1A ð5bÞ

LBD ¼ LBB1 �2 R cos αð Þ� �2

ð5cÞ

The coordination of points A, E and D, which are shown inFig. 4, are as follows:

A : ða1 cos θ;0;0ÞE : ð2R sin θ;2R cos θþa1 sin θ;0ÞD : ða1 cos θ;Rð1þ sin αÞ;0Þ

8><>: ð5dÞ

The slope of AE is:

tan β1 ¼ AbsoluteYE�YA

XE�XA

� �ð5eÞ

The coordination of point A″0 is:

A‴ : XEþYE�YD

tan β1; YD;0

� �ð5fÞ

LDA‴ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðXD�XA‴Þ2þðYD�YA‴Þ2

qð5gÞ

LDA″ ¼ LDA‴ sin β1 ð5hÞThe forth energy component is associated with the hinge lines

AB and AB1 as:

W4 ¼WABþWAB1 ¼ 2WAB ¼ 2Z

mpLABLBDr1ðLABÞ2

dl

¼ 2mpLBDr1ðLABÞ2

ðLABÞ33

" #¼ 2 mpLBDLAB

3r1ð6Þ

Fig. 3. Failure mode of steel hollow section under pure bending (Poonaya et al.[20]).

Fig. 4. YLM model for steel hollow sections (a) 3D view (b) front view.

S. Maduliat et al. / Thin-Walled Structures 88 (2015) 41–47 43

r1 ¼0:07�θ

70

� �a1; a1 ¼

132R ð6aÞ

LAB ¼2π R�tð Þð Þ�LBB1

2

� �ð6bÞ

Finally, the fifth and sixth energy components related to hingelines BF, B1F, BE and B1E as:

W5 ¼WBF þWBE ¼ 2AreaðBEDÞmp

r1; W6 ¼WB1FþWB1E ¼ 2AreaðB1ED1Þ

mp

r1ð7Þ

AreaðBEDÞ ¼ AreaðB1ED1Þ ¼ LBDa12

ð7aÞ

mp ¼Fyt2

4; ð8Þ

where Fy and t are the yield stress and the thickness of the section,respectively.

2.2. Estimating the ultimate moment capacity

The intersection of the failure curve and the elastic curves ofthe effective sections represent the ultimate capacity of thesections. In order to draw the failure curve for non-compact andcompact sections, the failure curves need to be shifted in order toproceed to the plastic stage. Therefore, the intersection of theelastic and failure curves prior to shifting of the failure curve willrepresent the ultimate capacity of the section (Fig. 5). A compar-ison of the pure bending real experiment results from the studiesby Elchalakani et al. [10] and Poonaya et al. [20] for bendingmoment capacities with ultimate moment capacities using YLManalysis are shown in Table 1. The ratio of the real experiment overthe YLM results are also shown in Table 1, with an average value of1.06 and coefficient of variation (COV) of 0.10.

After calculating the ultimate capacity using the YLM method,the shift in the failure curve warrants discussion. Compact sectionswill not fail before yielding therefore a plastic hinge develops as aresult. To classify a section as compact (plastic), not only should aplastic hinge develop at the maximum moment point, but also theplastic hinge should rotate sufficiently to redistribute the momentthrough the member. The shift in the failure curve depends on therotation capacity of the section; therefore, the concept of rotationcapacity (R), which is an indicative parameter of the section’sductility and determines how an internal moment can redistributewhen the plastic moment is reached, is important. It is assumedthat the rotation capacity varies from one to four for non-compactsections, and exceeds four for compact sections. Bambach et al. [5]have drawn on test results to propose a relationship between therotation capacity of a hat section and its slenderness value. In thisstudy, the experimental results based the research by Poonayaet al. [20] are used to generate an empirical equation for determin-ing the rotation capacity of a steel hollow section from itsslenderness ratio value.

λsrλsp; R¼ λspλs

� �4 atM ¼Mp ð9Þ

λspoλsrλsy; R¼ λsy�λsλsy�λsp

� �4 atM¼My ð10Þ

λsyoλs R¼ 0 ð11Þ

where Mp and My are plastic and yield moments, respectively; andλs is the value of the slenderness ratio and is calculated usingfollowing equation:

λs ¼d0t

� �Fy250

� �; ð12Þ

where d0 is the outside diameter of the section; and λsy and λsp areelastic and plastic slenderness limits, respectively, which are givenin AS4100 [1] as:

λsy ¼ 120λsp ¼ 50 ð13Þ

After calculating the rotation capacity of each sample, itsshifted moment–rotation curve can be determined using thefollowing equations:

Rotation¼ 0:5Le∅; ð14Þ

Intersection of the elastic and failure curves

Failure curve

Sample B1

Sample UB1

Rotation (deg)

Mom

ent (

kNm

)

Rotation (deg)

Mom

ent (

kNm

)

Fig. 5. Comparison between the test and the YLM diagram for: (a) slender section(b) compact section.

Table 1Comparing the test results with the YLM results.

Section R(mm)

t(mm)

Fy(MPa)

E (elastic)(MPa)

MYLM

(kN m)MTest

(kN m)D/t MTest/

MYLM

Poonaya et al. [20]UB1 29.6 2.8 330 128,000 2.98 3.20 21.16 1.07UB2 29.5 2.3 270 160,000 1.53 1.85 25.65 1.21UB3 23.4 1.8 320 173,000 1.20 1.12 26.03 0.93UB4 29.7 1.8 354 178,000 1.84 1.76 32.97 0.96UB5 29.3 1.6 257 128,000 1.09 1.09 36.59 1.00UB6 37.3 1.8 306 133,000 2.25 2.38 42.57 1.06

Elchalakani et al. [10]B1 55.1 1.1 408 190,900 3.91 3.90 100.09 1.00B2 55.0 1.0 408 190,900 3.29 3.80 109.90 1.16B3 54.9 0.9 408 190,900 2.80 3.39 121.89 1.21B4 55.2 1.3 408 190,900 4.65 4.50 88.32 0.97

Mean¼ 1.06Cov¼ 0.10

S. Maduliat et al. / Thin-Walled Structures 88 (2015) 41–4744

where Le is the effective length of the section, and ∅ is thecurvature of the section which is equal to:

∅¼ ðRþ1Þ∅p; ð15Þ

∅p is the plastic curvature, which can be calculated as follows:

∅p ¼Mp

EIð16Þ

where E and I are Young’s modulus of elasticity and secondmoment of area of a cross section, respectively.

Fig. 5 is a sample comparison between the experiment and theYLM diagram for the tested sections from Elchalakani et al. [10]and Poonaya et al. [20]. In Fig. 5, it is shown that compared to thereal experiment graph, the YLM graph for slender and shiftedcollapse curves for compact and non-compact sections is in goodagreement with the real experiment result.

Since the emphasis of this paper is on the collapse behaviour ofsteel hollow sections, the amount of absorbed energy due todeformation of the tested sections need to be measured; this isthe area under the moment–rotation curve.

3. Verifying the finite element model using the YLM model

The experimental method can be used to determine thecollapse behaviour and, therefore, the energy absorption of steelbollards under lateral impact load. This method is standardised inPAS 68 [21], but requires a significant amount of resources since itusually ends with the destruction of the vehicle and the bollards.

Numerical simulation of vehicle-to-bollard collision using finiteelement software is a good complementary approach.

As observed previously, the failure deformation of the bollardsafter collapse due to the lateral impact load (Fig. 2) and bendingload (Fig. 3) will follow a similar pattern. Therefore, in thisexercise, the FEM analysis approach is verified against the YLMresult, which has been verified against existing experimental data.

3.1. Material modelling

The Johnson–Cook material model, which is an empirical model,is used to simulate steel tube behaviour in the FEM [14]. It is to benoted that the Johnson–Cook model is an accurate and efficientmodel to define the mechanical properties of steel materials underdynamic events with high strain rate (Yuen and Nurick [23]).

The Johnson–Cook strength model can be expressed as:

σ ¼ AþBεn� �½1þC ln _εn�; ð17Þ

where ε is the equivalent plastic strain; _εn ¼ _ε=_ε0 is the dimension-less plastic strain for _ε¼ 1 s�1; and A, B, C and n are defined as partof the material definition. 0.2% of proof stress is used as constant A;and the strain hardening effects of the material are signified byconstants B and n. The strain rate effect is represented by constantC. According to Ding et al. [9] the A, B, C and n values for mild steelare 286 MPa, 500 MPa, 0.0171; and 0.228, respectively.

Fig. 6. Numerical simulation of a pick-up truck impacting on a bollard.

Fig. 7. Local and global deformation of the bollard subjected to a slow vehicleimpact.

0

50

100

150

200

250

300

350

0.00 0.15 0.30 0.45 0.60 0.75 0.90

Mom

ent (

kNm

)

Rotation (Rad)

200 mm dia x 20 mm thick bollard

FEMpre failurepost failure

0

200

400

600

800

1,000

1,200

0.00 0.05 0.10 0.15 0.20 0.25

Mom

ent (

kNm

)

Rotation (Rad)

320 mm dia x 20mm thick bollard

post failureFEMpre failure

Fig. 8. Moment–rotation diagram for (a) 200 mm dia bollard (b) 320 mm diabollard.

S. Maduliat et al. / Thin-Walled Structures 88 (2015) 41–47 45

3.2. Vehicle impact modelling

In this paper, a bollard with 20 mmwall thickness and 200 mmdiameters is modelled using the LS-DYNA finite element analysispackage. A vehicle crashed into the bollard with kinetic energy of612 kJ (Fig. 6). The vehicle model is obtained from a National CrashAnalysis Centre (NCAC) publication and is a typical representationof a 2500 kg vehicle load with the nominal lateral impact velocityof 20 mph (PAS 68 [21]).

Fig. 7 shows the typical local and global deformation of thebollard in response to the vehicle impacting it with kinetic energyof 612 kJ. It can also be seen in this figure that the failuremechanism for steel hollow sections based on the FEM and theexperimental observation are the same. Therefore, the verifiedYLM model can be used to validate the FEM.

Two local failure mechanisms of the bollard are observed inFig. 7. The failure mechanism at the top of the bollard is localdeformation due to the lateral impact load, whilst the mechanismat the bottom is due to the compression zone of the bollard. Theenergy that has been dissipated to form the failure mechanism atthe bottom of the bollard is equal to the work performed due tothe global movement of the bollard. Therefore, by establishing theYLM model for the sections and calculating the energy absorptionof each hinge line, the total absorbed energy can be estimated andthe failure curve can be plotted.

4. Comparing the FEM results with the YLM results

The moment–rotation graph of the modelled bollard is com-pared with the results based on the YLM model. The moment–rotation diagram for two modelled bollards is shown in Fig. 8. This

figure shows that the FEM graphs are in good agreement with theYLM results for a 320 mm diameter bollard.

As shown in Fig. 8, the moment–rotation diagram based on theFEM, for the 200 mm diameter bollard does not match well withthe YLM in the elastic range. This is due to local deformation of thebollard at the impacting location, which is explained in moredetail in the following section.

The effective stress versus rotation diagram is shown in Fig. 9and the two different critical locations of the bollard are pointedout in Fig. 10. As shown in Fig. 9, the stress at the location wherethe vehicle crashed into the bollard increased suddenly. Thismeans that the vehicle’s kinetic energy at the early stages ofcollision is mostly absorbed due to the local deformation of thebollard. By decreasing the bollards diameter from 320 mm to200 mm, the width-to-thickness ratio, and therefore the slender-ness ratio, of the bollard will drop. Thus, the section will be lessprone to local buckling and a greater load is needed to deform thebollard locally.

The FEM versus YLM graph for the 200 mm diameter bollardhas been compared using the hypothesis test technique to checkwhether the two sets of measurements are essentially different.The distribution of the difference is weakly normal. Therefore, inthis study, the Wilcoxon Signed Rank Test is used. The sectiongraph has been checked with 95% confidence interval for thedifferences, and the p-value is 0.21. The p-value is greater than0.05 and the mean of its difference includes zero within the 95%confidence interval. Therefore it is failed to reject two sets of dataare equal and cannot be seen that these two sets of data aresignificantly different.

It is noted that the indicative parameter for the section’sductility (rotation capacity) based on the FEM results is similarto the validated YLM results. This means that the section will fail ata same point in both methods. The modelled FEM in this study is adynamic simulation and the bollard will bounce back due to thedynamic response of the section to the lateral impact load.Therefore, to analyse the post-collapse response of the section,the YLM model can be used as a more reliable method.

5. Energy absorption of the bollard

Since the emphasis of this paper is on the collapse behaviour ofthe bollard under lateral impact load, the amount of absorbedenergy due to the deformation of the modelled sections isdetermined using the following equation:

E¼Z θ

0Mdθ ð18Þ

The area under the moment–rotation curve represents thedissipated energy. The Simpson rule, which is a method tocalculate the area under a graph based on the YLM method, isused to calculate the absorbed energy.

To measure the absorbed energy by the bollards in each FEMsimulation, the internal energy is used. Table 2 shows the values ofenergy absorbed by bollards of various sizes.

Table 2 compares the absorbed energy for the tested sectionsbased on the FEM and YLM results. From this table, it is evidentthat the ratio of energy absorption based on the YLM results overthe FEM results varies between 0.97 and 1.07. As highlighted inTable 2, the energy absorption based on the FEM is in a goodagreement with the YLM results.

A key point, identified during this study is that the energyabsorption of the section increases with decreasing width to thicknessratio. Table 2 shows that changing the size of the bollards, does notdramatically change their energy absorption due to global deforma-tion. However, the bollard size, and therefore its slenderness ratio has

0

200

400

600

800

0 0.05 0.1 0.15 0.2

Effe

ctiv

e st

ress

(MPa

)

Rotation (Rad)

200 dia x 20 mm thick bollard

local stressglobal stress

Fig. 9. Effective stress-rotation angle diagram.

Maximum effective stress due to global bending moment

Maximum effective stress due to local impact load

Fig. 10. Critical locations on the bollards.

S. Maduliat et al. / Thin-Walled Structures 88 (2015) 41–4746

a significant effect on its energy absorption due to local deformation.This is due to the fact that sections with smaller slenderness ratios areless vulnerable to local buckling (Guler et al. [11]). Therefore thesesections will not fail suddenly due to local buckling, and their strengthwill be controlled by material yielding prior to buckling failure.

In the recent work by Sun and Packer [22], authors have examinedthe strain-rate (from 100 s�1 to 1000 s�1) effects on a rectangularhollow section (RHS) and concluded that the RHS is not susceptible tolocal plastic failure due to the increase in strain-rate. This increase theloading rate will increase the ultimate load capacity of a section. It isalso noticed that loading velocity is less effective for rectangularhollow sections with higher yield capacity. In other research, Bambach[4] has investigated the behaviour of steel tubular columns subjectedto the blast load, which generate a (7500 m/s) impact wave on thestructure. In this work, he observed that local deformation of a slendersection is less susceptible to the blast loading compare to the compactsections. This could be attributed to the increase in yield strength dueto the increase in loading rate and the minimisation of dislocationsdue to the thin wall. The works by Kotelko and Mania [17] haveshowed that the failure curve will not be considerably affected byincreasing the loading velocity of up to 300 mm/s. In this work, wehave also compared simulation results of the 20 mph-impact on thesteel hollow tube with rate-dependent and rate-independent options.The differences observed in local and global responses in both eventsare, however, small confirming the negligible influences of loadingrate. The influence of impact velocity on failure behaviour of the steeltubes could be examined for higher impact velocities. It is recom-mended that there be further experimental and numerical analysis onthe behaviour of steel tube sections under different velocity impactloads and taking into account the change in strain rate during thecrushing process.

6. Conclusions

Yield Line Mechanism is an analytical technique that provides aless costly method to simulate the collapse response of thin-walled sections. This paper proposed a YLM model for steel hollowsections. After proposing the YLM model, collapse curves for eachexisting tested section are plotted using the energy method. Theultimate moment capacities of the tested samples are thendetermined using elastic and failure curves. It has been verifiedthat this model can be used for determining failure behavior ofsection. Shifting the failure curve is discussed for compact andnon-compact sections. Since the shift in the failure curve dependson the rotation capacity of the section, a method is proposed todetermine the rotation capacity for cold-formed channel sectionsunder bending. This enabled conclusions to be drawn that the YLMcollapse curves are in good agreement with the real experimentgraphs. Therefore, this model can be used to verify the FEM model.

Energy absorption due to failure based on the FEM results and YLMresults are compared. The ratios of YLM over FEM results for energyabsorption are between 0.97 and 1.07. The energy absorption of thesections can be increased by decreasing its width-to-thickness ratio.

By changing the size of the bollards, their energy absorption due toglobal deformation does not change dramatically. However, it has asignificant effect on its energy absorption due to local deformation. Itis to be noted that the strength of less slender sections will becontrolled by material yielding prior to local buckling failure.

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Table 2Section properties and energy absorption of the bollards.

Sample Re (mm) t (mm) λs h (mm) A (MPa) B (MPa) n C Energy EYLM (kJ) EFEM (kJ) EYLM/EFEM

Local (kJ) Global (kJ)

B32020 160 20 29 1000 286 500 0.228 0.0171 287 177 464 478 0.97B20020 100 20 18 1000 286 500 0.228 0.0171 442 223 666 621 1.07B32010 160 10 59 1000 286 500 0.228 0.0171 139 191 330 330 1.00

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